Towards Integrated Glance To Restructuring in Combinatorial Optimization

The paper focuses on a new class of combinatorial problems which consists in restructuring of solutions (as sets/structures) in combinatorial optimization. Two main features of the restructuring process are examined: (i) a cost of the restructuring, (ii) a closeness to a goal solution. Three types of the restructuring problems are under study: (a) one-stage structuring, (b) multi-stage structuring, and (c) structuring over changed element set. One-criterion and multicriteria problem formulations can be considered. The restructuring problems correspond to redesign (improvement, upgrade) of modular systems or solutions. The restructuring approach is described and illustrated (problem statements, solving schemes, examples) for the following combinatorial optimization problems: knapsack problem, multiple choice problem, assignment problem, spanning tree problems, clustering problem, multicriteria ranking (sorting) problem, morphological clique problem. Numerical examples illustrate the restructuring problems and solving schemes.Comment: 31 pages, 34 figures, 10 table

Towards Decision Support Technology Platform for Modular Systems

The survey methodological paper addresses a glance to a general decision support platform technology for modular systems (modular/composite alterantives/solutions) in various applied domains. The decision support platform consists of seven basic combinatorial engineering frameworks (system synthesis, system modeling, evaluation, detection of bottleneck, improvement/extension, multistage design, combinatorial evolution and forecasting). The decision support platform is based on decision support procedures (e.g., multicriteria selection/sorting, clustering), combinatorial optimization problems (e.g., knapsack, multiple choice problem, clique, assignment/allocation, covering, spanning trees), and their combinations. The following is described: (1) general scheme of the decision support platform technology; (2) brief descriptions of modular (composite) systems (or composite alternatives); (3) trends in moving from chocie/selection of alternatives to processing of composite alternatives which correspond to hierarchical modular products/systems; (4) scheme of resource requirements (i.e., human, information-computer); and (5) basic combinatorial engineering frameworks and their applications in various domains.Comment: 10 pages, 9 figures, 2 table

Online Multistage Subset Maximization Problems

Numerous combinatorial optimization problems (knapsack, maximum-weight matching, etc.) can be expressed as subset maximization problems: One is given a ground set N={1,...,n}, a collection F subseteq 2^N of subsets thereof such that the empty set is in F, and an objective (profit) function p: F -> R_+. The task is to choose a set S in F that maximizes p(S). We consider the multistage version (Eisenstat et al., Gupta et al., both ICALP 2014) of such problems: The profit function p_t (and possibly the set of feasible solutions F_t) may change over time. Since in many applications changing the solution is costly, the task becomes to find a sequence of solutions that optimizes the trade-off between good per-time solutions and stable solutions taking into account an additional similarity bonus. As similarity measure for two consecutive solutions, we consider either the size of the intersection of the two solutions or the difference of n and the Hamming distance between the two characteristic vectors. We study multistage subset maximization problems in the online setting, that is, p_t (along with possibly F_t) only arrive one by one and, upon such an arrival, the online algorithm has to output the corresponding solution without knowledge of the future. We develop general techniques for online multistage subset maximization and thereby characterize those models (given by the type of data evolution and the type of similarity measure) that admit a constant-competitive online algorithm. When no constant competitive ratio is possible, we employ lookahead to circumvent this issue. When a constant competitive ratio is possible, we provide almost matching lower and upper bounds on the best achievable one

Towards combinatorial clustering: preliminary research survey

The paper describes clustering problems from the combinatorial viewpoint. A brief systemic survey is presented including the following: (i) basic clustering problems (e.g., classification, clustering, sorting, clustering with an order over cluster), (ii) basic approaches to assessment of objects and object proximities (i.e., scales, comparison, aggregation issues), (iii) basic approaches to evaluation of local quality characteristics for clusters and total quality characteristics for clustering solutions, (iv) clustering as multicriteria optimization problem, (v) generalized modular clustering framework, (vi) basic clustering models/methods (e.g., hierarchical clustering, k-means clustering, minimum spanning tree based clustering, clustering as assignment, detection of clisue/quasi-clique based clustering, correlation clustering, network communities based clustering), Special attention is targeted to formulation of clustering as multicriteria optimization models. Combinatorial optimization models are used as auxiliary problems (e.g., assignment, partitioning, knapsack problem, multiple choice problem, morphological clique problem, searching for consensus/median for structures). Numerical examples illustrate problem formulations, solving methods, and applications. The material can be used as follows: (a) a research survey, (b) a fundamental for designing the structure/architecture of composite modular clustering software, (c) a bibliography reference collection, and (d) a tutorial.Comment: 102 pages, 66 figures, 67 table

Towards Electronic Shopping of Composite Product

In the paper, frameworks for electronic shopping of composite (modular) products are described: (a) multicriteria selection (product is considered as a whole system, it is a traditional approach), (b) combinatorial synthesis (composition) of the product from its components, (c) aggregation of the product from several selected products/prototypes. The following product model is examined: (i) general tree-like structure, (ii) set of system parts/components (leaf nodes), (iii) design alternatives (DAs) for each component, (iv) ordinal priorities for DAs, and (v) estimates of compatibility between DAs for different components. The combinatorial synthesis is realized as morphological design of a composite (modular) product or an extended composite product (e.g., product and support services as financial instruments). Here the solving process is based on Hierarchical Morphological Multicriteria Design (HMMD): (i) multicriteria selection of alternatives for system parts, (ii) composing the selected alternatives into a resultant combination (while taking into account ordinal quality of the alternatives above and their compatibility). The aggregation framework is based on consideration of aggregation procedures, for example: (i) addition procedure: design of a products substructure or an extended substructure ('kernel') and addition of elements, and (ii) design procedure: design of the composite solution based on all elements of product superstructure. Applied numerical examples (e.g., composite product, extended composite product, product repair plan, and product trajectory) illustrate the proposed approaches.Comment: 10 pages, 20 figures, 17 table

Toward breaking the curse of dimensionality: an FPTAS for stochastic dynamic programs with multidimensional actions and scalar states

We propose a Fully Polynomial-Time Approximation Scheme (FPTAS) for stochastic dynamic programs with multidimensional action, scalar state, convex costs and linear state transition function. The action spaces are polyhedral and described by parametric linear programs. This type of problems finds applications in the area of optimal planning under uncertainty, and can be thought of as the problem of optimally managing a single non-discrete resource over a finite time horizon. We show that under a value oracle model for the cost functions this result for one-dimensional state space is "best possible", because a similar dynamic programming model with two-dimensional state space does not admit a PTAS. The FPTAS relies on the solution of polynomial-sized linear programs to recursively compute an approximation of the value function at each stage. Our paper enlarges the class of dynamic programs that admit an FPTAS by showing, under suitable conditions, how to deal with multidimensional action spaces and with vectors of continuous random variables with bounded support. These results bring us one step closer to overcoming the curse of dimensionality of dynamic programming.Comment: Includes appendi

Efficient Algorithm for Nonpoint Source Pollution Control Problems

A dynamic programming algorithm is proposed for a class of nonpoint source pollution control problems. The inherently combinatorial nature of these problems--stemming from the discrete nature of the decision variables, which are production and conservation practices--gives them a special knapsack structure with multiple right hand sides and additional multiple choice constraints. This paper focuses on the computer implementation of this algorithm and its numerical testing and behavior compared with standard integer programming codes. The results show the robustness and relative efficiency of the approach. Furthermore, this paper demonstrates that dynamic programming can be used to generate sensitivity analysis information for multiple choice knapsack problems

A Graph Theoretic Approach to Non-Anticipativity Constraint Generation in Multistage Stochastic Programs with Incomplete Scenario Sets

We propose an algorithm for generating a minimum-cardinality set of non-anticipativity constraints (NAC) for scenario-based multistage-stochastic programming (MSSP) problems with both endogenous and exogenous uncertainties which allow for gradual realizations. Recently several authors have considered approaches to generate the minimum cardinality NAC set for MSSPs for various scenario set structures. However, these approaches have been limited to uncertain parameters where the realizations occur instantaneously or the full set of scenarios is required. The proposed algorithm, referred to as Sample Non-Anticipativity Constraint algorithm (SNAC) relaxes this requirement. We show that as long as the number of uncertain parameters and parameter values are kept constant, the algorithm scales polynomially in the number of scenarios

Two-Stage Memory Allocation using AHP & Knapsack at PT Berca Hardayaperkasa

We propose to manage a (MicroStrategy) Business Intelligence Server in terms of RAM allocation for its Intelligent Cubes as a two-stage resource allocation problem in which the first stage is formulated as an multi-criteria problem that can be solved using Analytic Hierarchy Process (AHP) and the second stage is multiple (several) 0-1 classic Knapsack problems with the constraints that are obtained using the result from the first stage. This Approach happens to have an advantage in terms of computational complexity as well, it reduces from O(nM) to O(max{nj}max{Mj}) when calculated in parallel. We illustrate our proposal with a numerical example based on our experience